Fubini
A131689 \(\bbox[yellow, 5px]{\color{DarkGreen} T_{n, k} \ = \ k! \ {n \brace k} } \)
A000007 \(T_{n , 0}\) ➤ TablCol0 ➤ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
A000012 \(T_{n + 1, 1}\) ➤ TablCol1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
---------- \(\sum_{k=0}^{n} T_{n, k}\ (-1)^{k}\) ➤ AltSum ➤ 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1
---------- \(\sum_{k=0}^{1} T_{1, n-k}\ n^k\) ➤ RevPolyRow1 ➤ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A000027 \(\sum_{k=0}^{1} T_{1, k}\ n^k\) ➤ PolyRow1 ➤ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
---------- \(\sum_{k=0}^{2} T_{2, n-k}\ n^k\) ➤ RevPolyRow2 ➤ 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
A000142 \(T_{n , n}\) ➤ TablDiag0 ➤ 1 1 2 6 24 120 720 5040 40320 362880
A000312 \(\sum_{k=0}^{n} T_{n, k} \ \binom{n}{k} \) ➤ BinConv ➤ 1 1 4 27 256 3125 46656 823543 16777216
A000629 \(\sum_{k=0}^{n} T_{n, n-k}\ (-2)^{n - k} \) ➤ RevNegHalf ➤ 1 -2 6 -26 150 -1082 9366 -94586
A000670 \(\sum_{k=0}^{n} T_{n, k}\) ➤ TablSum ➤ 1 1 3 13 75 541 4683 47293 545835
---------- \(\sum_{k=0}^{n} | T_{n, k} |\) ➤ AbsSum ➤ 1 1 3 13 75 541 4683 47293 545835
A000918 \(T_{n + 2, 2}\) ➤ TablCol2 ➤ 2 6 14 30 62 126 254 510 1022 2046 4094
A001117 \(T_{n + 3, 3}\) ➤ TablCol3 ➤ 6 36 150 540 1806 5796 18150 55980
A001286 \(T_{n + 1, n}\) ➤ TablDiag1 ➤ 0 1 6 36 240 1800 15120 141120 1451520
A002869 \(\text{max}_{k=0}^{n}\ | T_{n, k} |\) ➤ TablMax ➤ 1 1 2 6 36 240 1800 16800 191520
A004123 \(\sum_{k=0}^{n} T_{n, k}\ 2^k\) ➤ PolyCol2 ➤ 1 2 10 74 730 9002 133210 2299754
A005649 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ AccRevSum ➤ 1 2 8 44 308 2612 25988 296564 3816548
---------- \(\sum_{k=0}^{n} T_{n, k} \ (k + 1)\) ➤ TransNat1 ➤ 1 2 8 44 308 2612 25988 296564 3816548
A009006 \(\sum_{k=0}^{n} T_{n, k} \ (-2)^{n - k} \) ➤ NegHalf ➤ 1 1 0 -2 0 16 0 -272 0 7936 0 -353792 0
A014105 \(\sum_{k=0}^{2} T_{2, k}\ n^k\) ➤ PolyRow2 ➤ 0 3 10 21 36 55 78 105 136 171 210 253
A019538 \(T_{n + 1, k + 1} \) ➤ Toff11 ➤ 1 1 2 1 6 6 1 14 36 24 1 30 150 240 120
A027760 \(\text{gcd}_{k=0}^{n}\ | T_{n, k} |\ \ (T_{n,k}>1)\) ➤ TablGcd ➤ 1 1 2 6 2 30 2 42 2 30 2 66 2 2730 2 6
A028872 \(\sum_{k=0}^{3} T_{3, n-k}\ n^k\) ➤ RevPolyRow3 ➤ 6 13 22 33 46 61 78 97 118 141 166 193
A032033 \(\sum_{k=0}^{n} T_{n, k}\ 3^k\) ➤ PolyCol3 ➤ 1 3 21 219 3045 52923 1103781 26857659
A032109 \(\sum_{k=0}^{n} T_{n, n-k}\ [2 | k]\) ➤ RevEvenSum ➤ 1 1 2 7 38 271 2342 23647 272918
A037960 \(T_{n + 2, n}\) ➤ TablDiag2 ➤ 0 1 14 150 1560 16800 191520 2328480
A037961 \(T_{n + 3, n}\) ➤ TablDiag3 ➤ 0 1 30 540 8400 126000 1905120 29635200
A052841 \(\sum_{k=0}^{n} T_{n, k}\ [2 | k]\) ➤ EvenSum ➤ 1 0 2 6 38 270 2342 23646 272918
A069321 \(\sum_{k=0}^{n} T_{n, k} \ k\) ➤ TransNat0 ➤ 0 1 5 31 233 2071 21305 249271 3270713
A083411 \(\sum_{k=0}^{n} T_{n, k} \ k^{2}\) ➤ TransSqrs ➤ 0 1 9 79 765 8311 100989 1362439
A089677 \(\sum_{k=0}^{n} T_{n, k}\ (1 - [2 | k])\) ➤ OddSum ➤ 0 1 1 7 37 271 2341 23647 272917
A090582 \(T_{n + 1, n - k + 1} \) ➤ Trev11 ➤ 1 2 1 6 6 1 24 36 14 1 120 240 150 30 1
A094420 \(\sum_{k=0}^{n} T_{n, k}\ n^k\) ➤ PolyDiag ➤ 1 1 10 219 8676 544505 49729758
A094421 \(\sum_{k=0}^{3} T_{3, k}\ n^k\) ➤ PolyRow3 ➤ 0 13 74 219 484 905 1518 2359 3464 4869
A105795 \(\sum_{k=0}^{n/2} T_{n - k, k}\) ➤ AntiDSum ➤ 1 0 1 1 3 7 21 67 237 907 3741 16507
A120368 \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, j}\) ➤ AccSum ➤ 1 1 4 21 142 1175 11476 129073 1641802
---------- \(\sum_{k=0}^{n} \sum_{j=0}^{k} T_{n, n - j}\) ➤ RevAccRevSum ➤ 1 1 4 21 142 1175 11476 129073 1641802
---------- \(\sum_{k=0}^{n} T_{n, n-k}\ (k + 1)\) ➤ RevTransNat1 ➤ 1 1 4 21 142 1175 11476 129073 1641802
A122704 \(\sum_{k=0}^{n} T_{n, k} \ 2^{n - k} \) ➤ PosHalf ➤ 1 1 4 22 160 1456 15904 202672 2951680
A131689 \(T_{n, k}\) ➤ Triangle ➤ 1 0 1 0 1 2 0 1 6 6 0 1 14 36 24 0 1 30
---------- \(T_{n, k}\ (-1)^{k}\) ➤ Talt ➤ 1 0 -1 0 -1 2 0 -1 6 -6 0 -1 14 -36 24
A210029 \(T_{2 n, n}\) ➤ CentralE ➤ 1 1 14 540 40824 5103000 953029440
A233734 \(T_{2 n + 1, n}\) ➤ RevCentralO ➤ 1 6 150 8400 834120 129230640
A255927 \(\sum_{k=0}^{n} T_{n, n-k}\ 3^k\) ➤ RevPolyCol3 ➤ 1 1 5 33 285 3081 40005 606033 10491885
A331690 \(\sum_{k=0}^{n} T_{n, n-k}\ n^k\) ➤ RevPolyDiag ➤ 1 1 4 33 456 9445 272448 10386817
A344053 \(\sum_{k=0}^{n} T_{n, k} \ (-1)^{n - k} \ \binom{n}{k} \) ➤ InvBinConv ➤ 1 1 0 -9 -40 125 3444 18571 -241872
A344392 \(T_{n - k, k} \ \ (k \le n/2)\) ➤ Tantidiag ➤ 1 0 0 1 0 1 0 1 2 0 1 6 0 1 14 6 0 1 30
A344397 \(T_{n, n / 2}\) ➤ ColMiddle ➤ 1 0 1 1 14 30 540 1806 40824 186480
A367392 \(T_{2 n + 1, n}\) ➤ CentralO ➤ 0 1 30 1806 186480 29607600 6711344640